Current techniques include frequency domain implementations of corrections that are specifically directed to processing frames of data, see, e.g., U.S. Pat. No. 7,541,958. In other current techniques, a linear M-periodic system can be represented in the frequency domain by M×M matrixes (where M is the number of time interleaved ADC channels in our case). In order to get such representation, the distinct sets of M normalized frequencies
  f  ,      f    +          1      M        ,  …  ⁢          ,      f    +                  M        -        1            M      are considered. Complex exponents with these frequencies constitute eigen-spaces that are invariant under linear periodic transformations (see, e.g., R. Shenoy, D. Burnside, T. Parks, “Linear Periodic Systems and Multirate Filter Design,” IEEE Transactions on Signal Processing, vol. 42, no. 9, pp. 2242-2256, 1994, and M. Vetterli, “A Theory of Multirate Filter Banks,” IEEE Transactions on Acoustics, Speech, and Signal Processing, Vols. ASSP-35, no. 3, pp. 356-372, March 1987). In other words, the response of a Time Interleaved Analog to Digital Converter (TIADC) system to the weighted sum of complex exponentials from the same eigen-space is a weighted sum of the same complex exponentials at the same frequencies. This principle can be used for sinewaves instead of complex exponentials. The corresponding set of frequencies for real sinewaves in U.S. Pat. No. 7,541,958 is referred to as a Family of Mutual Alias Frequencies (FMAF). FMAF frequencies may be defined using the following notations: FS—the combined sampling rate of TIADC, M—number of time interleaved ADC channels, and P—an integer multiple of M which specifies frequency grid fp, according to the following:
                                          f            p                    =                                                    F                S                                            2                ⁢                                                                  ⁢                P                                      ⁢            p                          ,                                  ⁢                  p          =          0                ,        1        ,        …        ⁢                                  ,        P                            (        1        )            
Then, the i-th frequency index (i=0,1, . . . , M−1) inside the
  k  -      th    ⁢                  ⁢          (                        k          =          1                ,        …        ⁢                                  ,                              P            M                    -          1                    )      FMAF can be defined as follows:
                                          p            i                    ⁡                      (            k            )                          =                  {                                                                                          k                    +                                          2                      ⁢                                              Pi                        M                                                                              ,                                                                                                  if                    ⁢                                                                                  ⁢                    i                                    <                                      M                    2                                                                                                                                                                  -                      k                                        +                                          2                      ⁢                                                                        P                          ⁡                                                      (                                                          i                              -                                                              M                                2                                                            +                              1                                                        )                                                                          M                                                                              ,                                                                                                  if                    ⁢                                                                                  ⁢                    i                                    >                                                            M                      2                                        -                    1                                                                                                          (        2        )            
It should be noted that TIADC input signals with frequencies from the same k-th FMAF have the feature that DFT (Discrete Fourier Transform) at individual ADC outputs will have a single spectral tone at k-th bin.
Frequency domain correction can be derived by matrix multiplication as follows:hk(cor)=(Hk)−1hk  (3)where hk represents the vector with the m-th component, hk(m) representing a DFT in the k-th frequency bin at m-th ADC output before correction, hk(cor) represents a corresponding vector after correction, and matrix (Hk)−1 represents the inverse of the matrix with components Hk(m,i) derived in a calibration stage per DFT computation in the k-th bin at the m-th ADC output when a single tone having i-th frequency from the k-th FMAF was used for calibration. An example of frequency domain correction according to (3) is shown in FIG. 1.
Other techniques are directed to time domain implementation and are suitable for continuous data application, see, e.g., U.S. Pat. No. 8,009,070, which requires inversion of matrixes of size L×L, where L represents the number of finite impulse response (FIR) filter taps.